It is possible to account for both solar and atmospheric neutrino oscillations in a three flavor oscillation framework.
The three neutrino flavor states, \(\nu _{\alpha }\)  (\(\alpha = e, \mu , \tau \) ), mix to form three mass eigenstates \(\nu _i\)  (\(i=1,2,3\) ) with masses \(m_i\) .
There are two independent mass squared differences.
Without loss of generality, we take them to be \(\Delta m^2_{21} = m^2_2 - m_1^2 \equiv \Delta m^2_{\rm sol}\)  and \(\Delta m^2_{31} = m^3_2 - m_1^2 \equiv \Delta m^2_{\rm atm}\) .
Since \(\Delta m^2_{\rm sol} \ll \Delta m^2_{\rm atm}\) , we have \(\Delta m^2_{32} \approx \Delta m^2_{31}\) .
The unitary mixing matrix (PMNS matrix), connecting \(\nu _{\alpha }\)  to \(\nu _i\) , is parametrized in terms of three mixing angles (\(\theta _{12}, \theta _{13}\)  and \(\theta _{23}\) ) and one CP violating phase \(\delta _{\rm CP}\) .
Given that \(\theta _{13} \ll 1\)  [1]}, it was shown that \(\theta _{12} \approx \theta _{\rm sol}\)  and \(\theta _{23} \approx \theta _{\rm atm}\)  [2]}.
Later experiments with man-made sources have confirmed the three oscillation picture and have done precision measurements of the mass squared differences and the mixing angles [3]}.
Present data fixes \(\Delta m^2_{21}\)  to be positive, but does not fix the sign of \(\Delta m^2_{31}\) .
The three unknowns of the three flavor oscillation paradigm are
